The following is a tentative syllabus for the course. This page will be regularly updated as we go forward.
Wk | Lectures | Topics | Remarks |
---|---|---|---|
1 | 9/12 | Review of the Riemann Integral | |
9/14 | $\sigma$-algebras and Borel sets | ||
2 | 9/17 | Measurable functions | |
9/19 | Simple functions | ||
9/21 | Measures | ||
3 | 9/24 | Integral of positive functions | |
9/26 | Integration of complex functions | ||
9/28 | Dominated Convergence, null sets | ||
4 | 10/01 | Outer measures | Not in text |
10/03 | Lebesgue measure on the real line | Not in text | |
10/05 | Extension of premeasures | Folland | |
5 | 10/08 | Product measures | Folland |
10/10 | Integration on products | Folland | |
10/12 | $L^1$-spaces | ||
6 | 10/15 | Complex and signed measures | |
10/15 | Midterm exam | ||
10/17 | Modes of convergence | ||
10/19 | The Radon-Nikodym Theorem | ||
7 | 10/22 | Holomorphic functions | |
10/24 | Curves and paths | ||
10/26 | Integrals on paths | ||
8 | 10/29 | Cauchy's Theorem for convex sets | |
10/31 | Analyticity | ||
11/01 | Zeroes and singularities (x-hour) | ||
11/02 | Cauchy estimates | ||
9 | 11/05 | Local behavior of holomorphic functions | |
11/07 | Chains, cycles, homology | ||
11/09 | Cauchy's Theorem, homotopy invariance | ||
10 | 11/12 | Meromorphic functions, the Residue Theorem | |
11/16 | Final exam |